How to unwrap a model in 3d viewport (not UV unwrap)
I may be mistaken but I think you've gone a complicated way. It is less trivial to make a plane follow a curve than the opposite.

I propose you start with a plane instead.


Create a plane Scale it on X axis by 10 
Enter edit mode (tab) and select the top and bottom edges
Subdivide by 40 (spacebar then search for Subdivide)
Enable proportional edit mode (see screenshot) and select two vertices
Move the upwards and while moving, use your mouse wheel to increase the proportional editing falloff
Exit edit mode and rotate the plane by 90° on X axis
Create a circle (shift+a, curves > circle) and scale it by 400% (that's just an example)
Make sure the circle is at coordinates 0, 0, 0
Select the plane and add a modifier called Curve 
In the modifier, select the circle You should now see the plane wrapping around the circle
Adjust the scale of the plane to do a full revolution (here scale at 12.56)


And with this you're done. You can then apply the Skin modifier to the plane to give some depth. You can duplicate the plane the circle and make a smaller inner tube. There are many possibilities this way.

UV texturing is pretty easy too.